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The endomorphism tower of a finite symmetric group

This paper demonstrates that the endomorphism tower of the finite symmetric group $\mathcal{S}_n$ (for $n \geq 7$ ) does not stabilize in finitely many steps by explicitly characterizing the structure of the second and third iterated endomorphism monoids and showing their groups of units remain isomorphic to $\mathcal{S}_n$ .

Original authors: Victoria Gould, Ambroise Grau, Marianne Johnson, Jamie Smith

Published 2026-06-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Victoria Gould, Ambroise Grau, Marianne Johnson, Jamie Smith

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a massive, complex machine made of gears, levers, and switches. In the world of mathematics, this machine is a group called the Symmetric Group ( $S_n$ ), which represents all the possible ways you can shuffle a deck of $n$ cards.

This paper is about what happens when you ask a very specific question about this machine: "If we build a new machine that controls the first one, and then a third machine that controls the second, and so on, does the system ever stop changing?"

Here is a breakdown of the paper's journey, using simple analogies.

1. The Concept: The "Tower of Control"

Think of the original group ( $S_n$ ) as a Master Chef who knows every possible way to arrange ingredients.

Level 0: The Master Chef ( $S_n$ ).
Level 1: We build a Supervisor ( $End_1$ ). This supervisor's only job is to watch the Chef and record every possible way the Chef could change their own recipe (an "endomorphism"). The Supervisor is a new, larger machine.
Level 2: We build a Grand Supervisor ( $End_2$ ). This machine watches the first Supervisor and records every way that machine could change its own rules.
Level 3: A Great-Grand Supervisor ( $End_3$ ), and so on.

This sequence is called the Endomorphism Tower.

2. The Big Question: Does it Ever Stop?

In the world of pure math, there is a famous question about "Automorphism Towers" (where you only look at machines that rearrange things perfectly without breaking them). For finite groups, it was known that this tower eventually stops growing; it hits a ceiling and stabilizes.

The authors asked: What happens if we look at the "Endomorphism Tower" (where machines can break or simplify things)?

The Bad News: They proved that for any finite machine that isn't completely trivial (like a single button), this tower never stops growing. It keeps getting bigger and more complex forever. You can never reach a point where the "Grand Supervisor" looks exactly the same as the "Great-Grand Supervisor."

3. The Specific Case: The Symmetric Group ( $S_n$ )

Since the tower never stops, the authors decided to look at the first few floors of the tower for the Symmetric Group (the card shuffler), specifically when you have 7 or more cards ( $n \ge 7$ ).

They wanted to see if, even though the tower keeps growing, the "Leaders" (the invertible parts, or the "Group of Units") of these new machines looked familiar.

Floor 0: The Leader is the Symmetric Group ( $S_n$ ).
Floor 1: The Leader of the first Supervisor is also the Symmetric Group ( $S_n$ ). (This was already known).
Floor 2: The authors did the heavy lifting here. They mapped out every single rule and interaction in the second Supervisor machine ( $End_2$ ). They found that even though this machine is incredibly complex and huge, its Leader is still the Symmetric Group ( $S_n$ ).
Floor 3: They went one step further. They looked at the third Supervisor ( $End_3$ ) and confirmed that its Leader is also the Symmetric Group ( $S_n$ ).

4. How They Did It (The Detective Work)

To prove this, the authors had to act like detectives. They didn't just guess; they had to map out the entire "Cayley Table" (a giant multiplication chart) for the second floor of the tower.

They used a few clever tricks:

The "Fingerprint" Trick: They looked at specific parts of the machine that couldn't be moved or changed by any internal rearrangement (called "characteristic subsets").
The "Center" Trick: They analyzed which parts of the machine were "central" (didn't get moved around by others).
The "Parity" Trick: They used the fact that card shuffles can be "even" or "odd" to separate different types of rules.

By proving that the only way to rearrange the rules of the second and third floors without breaking them is to use the original Symmetric Group, they showed that the "soul" of the machine remains $S_n$ at these levels.

5. The Conclusion and Open Questions

The paper concludes with three main takeaways:

Infinite Growth: The tower of endomorphisms for a finite group never stabilizes; it grows forever.
Stable Leadership: Despite the tower growing forever, the "Leaders" (the groups of units) for the first three levels ( $End_0, End_1, End_2, End_3$ ) are all identical to the original Symmetric Group ( $S_n$ ).
The Mystery Continues: The authors ask the big question: Does this pattern hold forever? Is the leader of the 100th floor still $S_n$ ? They don't know yet. They also ask if there are other types of machines (monoids) where this tower does stop growing.

In short: The authors built a mathematical skyscraper of "control machines" based on card shuffling. They proved the building never stops getting taller, but they discovered that for the first few floors, the "CEO" of the building is always the same person: the original Symmetric Group.

Technical Summary: The Endomorphism Tower of a Finite Symmetric Group

Problem Statement
This paper investigates the "endomorphism tower" of a monoid $M$ , defined as the sequence of monoids $\text{End}_i(M)$ where $\text{End}_0(M) = M$ and $\text{End}_i(M) = \text{End}(\text{End}_{i-1}(M))$ for $i \ge 1$ . While the behavior of the analogous automorphism tower for finite groups is well-understood (terminating in finite steps for finite centerless groups), the behavior of the endomorphism tower for monoids remains largely unexplored. The authors first establish that for any finite non-trivial monoid, the sequence $\text{End}_i(M)$ never stabilizes (i.e., $\text{End}_i(M) \not\cong \text{End}_{i+1}(M)$ for any finite $i$ ). Consequently, the study shifts to identifying stability patterns in the groups of units (automorphism groups) of these towers. The primary focus is the finite symmetric group $S_n$ for $n \ge 7$ , where the automorphism tower terminates immediately ( $\text{Aut}(S_n) \cong S_n$ ). The central question is: does the group of units of $\text{End}_i(S_n)$ remain isomorphic to $S_n$ for higher levels $i$ ?

Methodology
The authors employ a combination of structural semigroup theory and specific properties of symmetric groups to analyze the monoids $\text{End}_1(S_n)$ , $\text{End}_2(S_n)$ , and $\text{End}_3(S_n)$ .

Structural Decomposition: The paper utilizes Green's relations and the classification of idempotents to partition the monoids. For $\text{End}_1(S_n)$ , the structure is recalled from prior literature, distinguishing between automorphisms, rank 2 singular endomorphisms (idempotent and non-idempotent), and the zero element.
Characteristic Subsets and Centralizers: A core technique involves identifying "characteristic subsets" (subsets invariant under all automorphisms) within the endomorphism monoids. The authors leverage the fact that for $n \ge 7$ , $S_n$ is a complete group (centerless with only inner automorphisms). This allows the decomposition of $\text{Aut}(\text{End}_i(S_n))$ into a direct product involving $S_n$ and a kernel of automorphisms fixing the group of units.
Centralizer Analysis of Order 2 Elements: A significant portion of the technical work (Sections 4.1 and 5) is dedicated to analyzing the centralizers of elements of order 2 in $S_n$ . The authors prove lemmas establishing that if two elements of order 2 have nested or equal centralizers, they must be equal or disjoint under specific parity conditions. This is crucial for distinguishing conjugacy classes within the singular endomorphisms.
Iterative Construction: The authors explicitly construct the elements of $\text{End}_2(S_n)$ by classifying them based on their action on the group of units $\text{Aut}(S_n)$ and the singular part $\text{SEnd}(S_n)$ . They categorize endomorphisms by rank and kernel structure, providing a complete multiplication table for $\text{End}_2(S_n)$ .
Automorphism Determination: To determine the group of units of $\text{End}_3(S_n)$ (which is $\text{Aut}(\text{End}_2(S_n))$ ), the authors systematically prove that any automorphism of $\text{End}_2(S_n)$ that fixes the group of units $\text{Aut}(S_n)$ pointwise must be the identity map. This is achieved by showing that such automorphisms must fix specific idempotents and characteristic subsets, eventually forcing the entire monoid to be fixed.

Key Contributions and Results

Non-Stabilization of Finite Monoids: The paper proves (Proposition 3.2) that the endomorphism tower of any finite non-trivial monoid never stabilizes in a finite number of steps, as the number of idempotents strictly increases at each step.
Structure of $\text{End}_2(S_n)$ : The authors provide a complete description of the elements of $\text{End}_2(S_n)$ for $n \ge 7$ . They classify these endomorphisms into distinct families (automorphisms, rank 2 maps, maps collapsing conjugacy classes, etc.) and present the full composition table (Section 6).
Isomorphism of Unit Groups:
- It is verified that the group of units of $\text{End}_1(S_n)$ is isomorphic to $S_n$ .
- The paper determines the structure of $\text{End}_2(S_n)$ and proves that its group of units, $\text{Aut}(\text{End}_1(S_n))$ , is isomorphic to $S_n$ .
- Crucially, the authors determine the structure of $\text{End}_3(S_n)$ and prove that its group of units, $\text{Aut}(\text{End}_2(S_n))$ , is also isomorphic to $S_n$ (Theorem 7.23).
Characterization of Automorphisms: The paper demonstrates that for $n \ge 7$ , the automorphism group of $\text{End}_2(S_n)$ consists only of inner automorphisms, mirroring the property of $S_n$ itself.

Significance and Claims
The authors position this work as an "opener" to the study of endomorphism towers of monoids. They acknowledge that while the automorphism tower of $S_n$ is known to stabilize at the first step, the behavior of the endomorphism tower above level $i=1$ was previously unknown.

The primary significance claimed is the determination of the elements and structure of $\text{End}_2(S_n)$ and the verification that the group of units remains $S_n$ for the first three levels of the tower ( $i=0, 1, 2$ ). The paper explicitly states that it illustrates the techniques required for such studies and notes the complexities that arise.

The authors remain modest regarding the general case. They pose the open question (Question 8.1) of whether $\text{Aut}(\text{End}_i(S_n)) \cong S_n$ for all $i \ge 3$ , noting that a general pattern for elements passing from one level to the next has not yet been discovered. They also frame the work as a foundation for studying other transformation monoids (like $T_n$ ) and for investigating whether infinite semigroups can possess stabilizing endomorphism towers.

1. The Concept: The "Tower of Control"

2. The Big Question: Does it Ever Stop?

3. The Specific Case: The Symmetric Group (SnS_nSn​)

4. How They Did It (The Detective Work)

5. The Conclusion and Open Questions

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3. The Specific Case: The Symmetric Group ( $S_n$ )